Dimensions and waiting times for Gibbs measures (Q1976748)

Let \(\Omega\) be a subshift of finite type with shift \(T\), \(\nu\) a \(T\)-invariant measure and \(\mu\) a Gibbs measure (with potential \(\varphi\)). It is shown that the set \(G(\nu)\) of \(\nu\)-generic points has Billingsley dimension \(\dim_\mu G(\nu)= {h(\nu)\over h(\nu)+ h(\nu\mid\mu)}\), where \(h(\nu|\mu)\) denotes the relative entropy. The main result states that for ergodic \(\nu\) the waiting time \[ W_n(x,y)= \inf\{k\geq 1: (T^k(y))_i= x_i\quad (1\leq i\leq n)\} \] satisfies \[ \lim_{n\to\infty} {1\over n}\log W_n= {h(\nu)\over \dim_\mu(G(\nu))}\qquad \nu\times \mu\text{-a.e.}. \]